@JB Welcome to Microsoft Q & A Community Forum. In regard to your request for an example on how to set a custom webhook payload within the UI, unfortunately this is not available as custom JSON-based webhooks are not supported within the latest API version, 2021-08-01, and this is the version that is used when creating new log alerts via the UI.
However, as you are already aware, you can still create the previous version of log alerts outside of the UI utilizing API version 2018-04-16 and configuring your custom JSON payload as shown in this document.
As a workaround, you can have an action group trigger a Logic app where you get the alert details through payload and then send an email notification as per your requirement. Refer here more details.
I also would recommend you navigate here and share your feedback or suggestions directly with the responsible Azure feature team and clicking the vote button of your suggestion to raise visibility and priority on it.